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A255518 Number of rooted identity trees with n nodes and 6-colored non-root nodes.

%I A255518 #12 Apr 13 2019 22:19:45
%S A255518 0,1,6,51,542,6273,77262,991264,13109394,177428496,2445799878,
%T A255518 34218909015,484668543660,6935979521095,100136840127534,
%U A255518 1456726481340477,21332234689262126,314211310198197978,4652073871301311128,69194159284034697860,1033441834436615884266
%N A255518 Number of rooted identity trees with n nodes and 6-colored non-root nodes.
%H A255518 Alois P. Heinz, <a href="/A255518/b255518.txt">Table of n, a(n) for n = 0..800</a>
%F A255518 a(n) ~ c * d^n / n^(3/2), where d = 16.1255293604485586705050971466317639696978222052989685416919877643317..., c = 0.065726585606326795496161701400246926800764944... . - _Vaclav Kotesovec_, Feb 24 2015
%F A255518 From _Ilya Gutkovskiy_, Apr 13 2019: (Start)
%F A255518 G.f. A(x) satisfies: A(x) = x*exp(6*Sum_{k>=1} (-1)^(k+1)*A(x^k)/k).
%F A255518 G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * Product_{n>=1} (1 + x^n)^(6*a(n)). (End)
%p A255518 with(numtheory):
%p A255518 a:= proc(n) option remember; `if`(n<2, n, -add(a(n-j)*add(
%p A255518       6*a(d)*d*(-1)^(j/d), d=divisors(j)), j=1..n-1)/(n-1))
%p A255518     end:
%p A255518 seq(a(n), n=0..30);
%Y A255518 Column k=6 of A255517.
%K A255518 nonn
%O A255518 0,3
%A A255518 _Alois P. Heinz_, Feb 24 2015